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Abstract: Geometry of Interaction is a transcendental syntax developed in the framework of operator algebras. This fifth installment of the program takes place inside a von Neumann algebra, the hyperfinite factor. It provides a built-in interpretation of cut-elimination as well as an explanation for light, i.e., complexity sensitive, logics. [Copyright &y& Elsevier]

Abstract: We give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped net. We first prove that: (1) a net is head-normalizable (i.e. normalizable at depth ) if and only if its interpretation in the multiset based relational semantics is not empty and (2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut-link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus. [Copyright &y& Elsevier]